# Tagged Questions

the problem of finding the path between two points such that the transit time under specified conditions is minimized.

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### Derivative condition in the Brachistochrone problem

I know that in general, to find the minimum of some line integral with given end points I need to solve E-L equations. what disturbs me is that I have seen in my class the famous Brachistochrone ...
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### Ball on a slope with hollow

The experiment shown in the image suggests that ball B will reach the goal faster than ball A although the balls have identical properties and they start from the same height. The authors even ...
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### What is the position as a function of time for a mass falling down a cycloid curve?

In the brachistochrone problem and in the tautochrone problem it is easy to see that a cycloid is the curve that satisfies both problems. If we consider $x$ the horizontal axis and $y$ the vertical ...
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### Brachistochrone parametric equations

I'm having a bit of a hard time understanding how the parametrized $y$ equation (given below) of the brachistochrone is correct. When these equations are plotted it gives a concave down graph. Shouldn'...
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### Comparing Brachistochrone curve with a Hypocycloid curve

I want to compare the time that it takes to slide a particle in a frictionless hypocycloid curve, so time would be given by the arclength divided by the velocity So I need first compute the ...
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### Optimal tunnel shape for travelling inside the earth [duplicate]

Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (...
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### Why is a cycloid path the fastest way to roll a ball downward? [duplicate]

Possible Duplicate: Path to obtain the shortest traveling time I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal ...
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### Other kind of Brachistochrone problem

We know the Brachistochrone problem that to find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the ...
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### What shape of track minimizes the time a ball takes between start and stop points of equal height?

I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end ...
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### Brachistochrone problem for 3 points

I wonder how I can solve the Brachistochrone problem for 3 points? The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in ...
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### Path to obtain the shortest traveling time

Asume we have a particle sitting at the point A(0,0) in a gravitational field. (g=9.81) It is going to move along some path to the point B(a,b) Where a>0 and b<0. What is the curve the particle ...
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### Is there an intuitive reason the brachistochrone and the tautochrone are the same curve?

The brachistochrone problem asks what shape a hill should be so a ball slides down in the least time. The tautochrone problem asks what shape yields an oscillation frequency that is independent of ...